25  Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a principle to find the best parameter for a probability distribution that best explains a sampled dataset. The instances in the dataset are supposed be independently sampled from the same distribution. MLE gives a relatively easy way to estimate parameters from the data.

Maximum likelihood

Likelihood function

Given the dataset \mathcal{X} = \{ \mathbf{x}_{1}, \dots, \mathbf{x}_{n} \}, the term likelihood function simply refers to the probability function of \mathcal{X} with the distribution parameters for \mathbf{X} being \boldsymbol{\theta}

f(\boldsymbol{\theta}) = \mathbb{P}_{\mathbf{X}}(\mathcal{X} ; \boldsymbol{\theta}),

which measures how likely are the parameters \boldsymbol{\theta} given the data \mathcal{X}.

Notes:

  • Since the dataset are known, the probability function is only dependent on the parameters \boldsymbol{\theta}, and thus doesn’t have the same shape as the probability density of the variable \mathbf{X}.

  • The semicolon indicate that \boldsymbol{\theta} is a parameter (fixed value) instead of a random variable.

MLE Procedures

  1. We collect a set of instances \mathcal{X} = \{ \mathbf{x}_{1}, \dots, \mathbf{x}_{n} \} that we believe should be from the same distribution.

  2. We select a parametric model \mathbb{P}_{\mathbf{X}}(\mathbf{x} ; \boldsymbol{\theta}) that we think can best explains the data.

  3. We select the parameters \boldsymbol{\theta}^{*} to be the ones that maximize the probability of the data:

\begin{aligned} \boldsymbol{\theta}^{*} & = \arg\max_{\boldsymbol{\theta}} \mathbb{P}_{\mathbf{X}}(\mathcal{X}; \boldsymbol{\theta}) \\ & = \arg\max_{\boldsymbol{\theta}} \prod_{i = 1}^{n} \mathbb{P}_{\mathbf{X}}(\mathbf{x}_{i} ; \boldsymbol{\theta}) & [\mathcal{X} \text{ are i.i.d samples }] \\ & = \arg\max_{\boldsymbol{\theta}} \ln \prod_{i = 1}^{n} \mathbb{P}_{\mathbf{X}}(\mathbf{x}_{i} ; \boldsymbol{\theta}) & [\text{the log trick}] \\ & = \arg\max_{\boldsymbol{\theta}} \sum_{i = 1}^{n} \log \mathbb{P}_{\mathbf{X}}(\mathbf{x}_{i} ; \boldsymbol{\theta}). \end{aligned}

Optimization methods

If the likelihood function is concave, the best parameters \boldsymbol{\theta}^{*} that maximize the likelihood function are the ones that make the gradient of f(\boldsymbol{\theta}) with respect to \boldsymbol{\theta} 0 and at the same time has negative semidefinite hessian matrix.

\nabla_{\boldsymbol{\theta}} f(\boldsymbol{\theta}) = \nabla_{\boldsymbol{\theta}} \mathbb{P}_{\mathbf{X}}(\mathcal{X} ; \boldsymbol{\theta}) = 0

\nabla_{\boldsymbol{\theta}}^{2} f(\boldsymbol{\theta}) \preceq 0

Otherwise, other numerical methods or algorithms might need to employed to solve the optimization problem.

Example: linear regression

Given an instance \mathbf{x} \in \mathbb{R}^{d}, the function f (\cdot) is a linear function if it has the form

\begin{aligned} f (\mathbf{x}) & = \mathbf{w}^{T} \mathbf{x} + b \\ & = \boldsymbol{\theta}^{T} \hat{\mathbf{x}}, \\ \end{aligned}

where

  • \mathbf{w} is weight vector and b is the bias term,

  • \boldsymbol{\theta} is the parameter vector that includes both weights and bias

    \boldsymbol{\theta} = [ \mathbf{w}, b ]^{T},

  • \hat{\mathbf{x}} is the instance vector appended with a constant 1

    \hat{\mathbf{x}} = [ \mathbf{x}, 1 ]^{T}.

Given a dataset with instances \mathcal{X} = \{ \mathbf{x}_{1}, \dots, \mathbf{x}_{n} \} and labels \mathcal{Y} = \{ y_{1}, \dots, y_{n} \}, linear regression is often formulated as a problem of finding parameters \boldsymbol{\theta} that minimize the mean squared error (MSE) loss function

\begin{aligned} & \arg\min_{\boldsymbol{\theta}} L_{\text{MSE}} \\ & = \arg\min_{\boldsymbol{\theta}} \frac{1}{n} \sum_{i=1}^{n} \left( y_{i} - \boldsymbol{\theta}^{T} \hat{\mathbf{x}} \right)^{2} \\ & = \arg\min_{\boldsymbol{\theta}} \sum_{i=1}^{n} \left( y_{i} - \boldsymbol{\theta}^{T} \hat{\mathbf{x}} \right)^{2} \\ \end{aligned}

which can be written in matrix notations by treating \mathcal{X} as a matrix \mathbf{X} \in \mathbb{R}^{n \times d} and \mathcal{Y} \in \mathbb{R}^{n \times 1} as a vector \mathbf{y}

\arg\min_{\boldsymbol{\theta}} \lVert \mathbf{y} - \hat{\mathbf{X}} \boldsymbol{\theta} \rVert^{2}_{2},

where \hat{\mathbf{X}} is the instance matrix append with a column of 1 in the last column.

Linear regression as MLE

Solving the optimization problem of linear regression with MSE loss can be formulated as solving MLE of parameters for a univariate normal distribution.

First we can consider the difference \epsilon between y and \boldsymbol{\theta}^{T} \mathbf{x} for any instance \mathbf{x} as a random variable that follows the a univariate normal distribution with zero mean and known variance

y - \boldsymbol{\theta}^{T} \hat{\mathbf{x}} = \epsilon \sim \mathcal{G} \left( \epsilon, 0, \sigma^{2} \right).

Since \boldsymbol{\theta}^{T} \mathbf{x} is a not a random variable (constant), the label y for any instance \mathbf{x} is thus a univariate normal random variable with its mean being the predicted label of the linear function

\begin{aligned} y & = \epsilon + \boldsymbol{\theta}^{T} \hat{\mathbf{x}} \\ & \sim \mathcal{G} \left( y, \boldsymbol{\theta}^{T} \hat{\mathbf{x}}, \sigma^{2} \right), \end{aligned}

In other words, the conditional probability of any label given the instance \mathbf{x} follows a univariate normal distribution

\mathbb{P}_{Y \mid \mathbf{X}} \left( y \mid \mathbf{x} \right) = \mathcal{G} \left( y, \boldsymbol{\theta}^{T} \hat{\mathbf{x}}, \sigma^{2} \right).

Given a set of instances \mathcal{X} and their labels \mathcal{Y}, \mathbb{P}_{Y \mid \mathbf{X}} \left( y \mid \mathbf{x} \right) is a likelihood function of parameters \boldsymbol{\theta} and thus MLE can be used to find the best parameters, which can be shown to have the optimization problem of fitting a linear function with MSE loss

\begin{aligned} \boldsymbol{\theta}^{*} & = \arg\max_{\boldsymbol{\theta}} \mathbb{P}_{Y \mid \mathbf{X}} \left( y \mid \mathbf{x} \right) \\ & = \arg\min_{\boldsymbol{\theta}} \sum_{i=1}^{n} \left( y_{i} - \boldsymbol{\theta}^{T} \hat{\mathbf{x}}_{i} \right)^{2} \\ & = \arg\min_{\boldsymbol{\theta}} \lVert \mathbf{y} - \hat{\mathbf{X}} \boldsymbol{\theta} \rVert^{2}_{2}. \end{aligned}

\begin{aligned} \boldsymbol{\theta}^{*} & = \arg\max_{\boldsymbol{\theta}} \mathbb{P}_{Y \mid \mathbf{X}} \left( y \mid \mathbf{x} \right) \\ & = \arg\max_{\boldsymbol{\theta}} \sum_{i=1}^{n} \log \mathcal{G} \left( y, \boldsymbol{\theta}^{T} \hat{\mathbf{x}}, \sigma^{2} \right) \\ & = \arg\max_{\boldsymbol{\theta}} \sum_{i=1}^{n} \log - \frac{ 1 }{ \sqrt{2 \pi \sigma^{2}} } \exp{ - \frac{ \left( y_{i} - \boldsymbol{\theta}^{T} \hat{\mathbf{x}}_{i} \right)^{2} }{ 2 \sigma^{2} }} \\ & = \arg\max_{\boldsymbol{\theta}} \sum_{i=1}^{n} - \frac{1}{2} \log \left( 2 \pi \sigma^{2} \right) - \sum_{i=1}^{n} \frac{ \left( y_{i} - \boldsymbol{\theta}^{T} \hat{\mathbf{x}}_{i} \right)^{2} }{ 2 \sigma^{2} } \\ & = \arg\max_{\boldsymbol{\theta}} - \sum_{i=1}^{n} \left( y_{i} - \boldsymbol{\theta}^{T} \hat{\mathbf{x}}_{i} \right)^{2} & [\text{only need term with } \boldsymbol{\theta}] \\ & = \arg\min_{\boldsymbol{\theta}} \sum_{i=1}^{n} \left( y_{i} - \boldsymbol{\theta}^{T} \hat{\mathbf{x}}_{i} \right)^{2} \\ \end{aligned}

Solving linear regression

Solving linear regression is the same in both formulations by setting the gradient w.r.t \boldsymbol{\theta} to 0 and solve for \boldsymbol{\theta}

\begin{aligned} \nabla_{\boldsymbol{\theta}} \lVert \mathbf{y} - \hat{\mathbf{X}} \boldsymbol{\theta} \lVert^{2}_{2} & = 0 \\ 2 \left( \hat{\mathbf{X}}^{T} \hat{\mathbf{X}} \boldsymbol{\theta} - \hat{\mathbf{X}}^{T} \mathbf{y} \right) & = 0 \\ \hat{\mathbf{X}}^{T} \hat{\mathbf{X}} \boldsymbol{\theta} & = \hat{\mathbf{X}}^{T} \mathbf{y} \\ \boldsymbol{\theta} & = \left( \hat{\mathbf{X}}^{T} \hat{\mathbf{X}} \right)^{-1} \hat{\mathbf{X}}^{T} \mathbf{y}. \\ \end{aligned}